1. Field of the Invention
The invention refers to a method for estimating hidden channel parameters, specifically the delay, the amplitude and the phase, of a GNSS navigation signal received in a dynamic multipath environment, using a sequential estimation by means of a recursive Bayesian filtering which starts from the likelihood value of the measured channel output signal and updates the value using a state transition model, wherein previous knowledge about the statistic dependences of subsequent sets of time-varying channel parameters is utilized by introducing a movement model approximated to the actual channel environment, which model corresponds to a Markov process and forms the state transition model, the knowledge that reflection signals typically have a life cycle starting from their first appearance and experiencing a gradual change in delay, amplitude and phase over time until they disappear, being utilized for an a-posteriori prediction, and, with this movement model as the basis, the channel parameters are estimated sequentially by means of the recursive Bayesian filter, the number of paths being implicitly included in the estimation, and the result of this estimation is not only a fixed estimated value, but yields a-posteriori probability density functions of the estimated channel parameters.
2. Description of Related Art
Measuring the propagation time of signals emitted from sources of known position to a receiver allows to determine the position of the receiver in three-dimensional space. As a precondition, the receiver in a synchronous system (synchronous receivers and sources) must be able to receive the signals from at least three sources. If the receiver clock is not synchronous with the clocks of the sources, the propagation time measurement, and thus the determination of position, requires signals from four sources, since the clock error of the receiver must be calculated in addition to the coordinates. This method of determining a position is referred to as “Time of Arrival” (TOA) or as “pseudo-ranging” and is employed in satellite navigation methods, for example.
With a two-dimensional problem, for example, the propagation time estimates form circles around the respective sources, the points of intersection of the circles representing the possible positions of the receivers. In this context, reference is made to FIG. 1, in which the sources 1 and 2, emitting radio waves, are provided and the circles around the two sources 1 and 2 are identified as 3 and 4. Up to the circle 3, the estimated propagation time is t1, and to the circle 4, the estimated propagation time is t2. one of the two points of intersection 5 and 6 between the two circles 3 and 4 can often be excluded on the basis of previous knowledge so that an unambiguous solution is obtained.
Analogously, in a three-dimensional case, the points of intersection are located on spheres whose centres form the sources.
Problematic for the TOA method are the errors in the respective propagation time estimates that take effect as errors in the estimation of position.
By measuring the individual signal propagation times τ1,j,k and by using the known positions of the sources [pj,kt,x pj,kt,y pj,kt,z] at the time k, the geometry for j=1, . . . , M sources yields the following non-linear equation system for the vector of the relevant receiver parameters p=[pkr,x pkr,y pkr,z τkr]T, where, pkr,x, pkr,y, pkr,z are the receiver coordinates and τkr is the clock error of the receiver.
                                                        c                              -                1                                      ⁢                                                                                                                                                                                                                                                  (                                                                                                      p                                    k                                                                          r                                      ,                                      x                                                                                                        -                                                                      p                                                                          1                                      ,                                      k                                                                                                              t                                      ,                                      x                                                                                                                                      )                                                            2                                                        +                                                                                                                                                                                                                                          (                                                                                                      p                                    k                                                                          r                                      ,                                      y                                                                                                        -                                                                      p                                                                          1                                      ,                                      k                                                                                                              t                                      ,                                      y                                                                                                                                      )                                                            2                                                        +                                                                                                                                                                                                                                  (                                                                              p                            k                                                          r                              ,                              z                                                                                -                                                      p                                                          1                              ,                              k                                                                                      t                              ,                              z                                                                                                      )                                            2                                                                                                    +                      τ            k            r                    -                      ɛ                          1              ,              k                                      =                  τ                      1            ,            1            ,            k                                                  ⋮        ⁢                                                                                              c                              -                1                                      ⁢                                                                                                                                                                                                                                                  (                                                                                                      p                                    k                                                                          r                                      ,                                      x                                                                                                        -                                                                      p                                                                          j                                      ,                                      k                                                                                                              t                                      ,                                      x                                                                                                                                      )                                                            2                                                        +                                                                                                                                                                                                                                          (                                                                                                      p                                    k                                                                          r                                      ,                                      y                                                                                                        -                                                                      p                                                                          j                                      ,                                      k                                                                                                              t                                      ,                                      y                                                                                                                                      )                                                            2                                                        +                                                                                                                                                                                                                                  (                                                                              p                            k                                                          r                              ,                              z                                                                                -                                                      p                                                          j                              ,                              k                                                                                      t                              ,                              z                                                                                                      )                                            2                                                                                                    +                      τ            k            r                    -                      ɛ                          j              ,              k                                      =                  τ                      1            ,            j            ,            k                                                  ⋮        ⁢                                                                                              c                              -                1                                      ⁢                                                                                                                                                                                                                                                  (                                                                                                      p                                    k                                                                          r                                      ,                                      x                                                                                                        -                                                                      p                                                                          M                                      ,                                      k                                                                                                              t                                      ,                                      x                                                                                                                                      )                                                            2                                                        +                                                                                                                                                                                                                                          (                                                                                                      p                                    k                                                                          r                                      ,                                      y                                                                                                        -                                                                      p                                                                          M                                      ,                                      k                                                                                                              t                                      ,                                      y                                                                                                                                      )                                                            2                                                        +                                                                                                                                                                                                                                  (                                                                              p                            k                                                          r                              ,                              z                                                                                -                                                      p                                                          M                              ,                              k                                                                                      t                              ,                              z                                                                                                      )                                            2                                                                                                    +                      τ            k            r                    -                      ɛ                          M              ,              k                                      =                  τ                      1            ,            M            ,            k                              
Here, c is the speed of the signal (speed of light) and the values of εj,k are correction terms assumed to be known (e.g. terms for the correction of ionosphere, troposphere and clocks). Since the system of equations is non-linear, the same is generally linearized and solved iteratively. If, summarizing,
                    H        j            ⁡              (        p        )              =                                        c                          -              1                                ⁢                                                                      (                                                            p                      k                                              r                        ,                        x                                                              -                                          p                                              j                        ,                        k                                                                    t                        ,                        x                                                                              )                                2                            +                                                (                                                            p                      k                                              r                        ,                        y                                                              -                                          p                                              j                        ,                        k                                                                    t                        ,                        y                                                                              )                                2                            +                                                (                                                            p                      k                                              r                        ,                        z                                                              -                                          p                                              j                        ,                        k                                                                    t                        ,                        z                                                                              )                                2                                                    +                  τ          k          r                -                  ɛ                      j            ,            k                              =              τ                  1          ,          j          ,          k                      ,then the following linearized system of equation is obtained
                    (                                                                                                  ⅆ                                                                  H                        1                                            ⁡                                              (                        p                        )                                                                                                  ⅆ                                          p                      k                                              r                        ,                        x                                                                                            ⁢                                  |                                      p                    =                                          p                      0                                                                                                                                                                ⅆ                                                                  H                        1                                            ⁡                                              (                        p                        )                                                                                                  ⅆ                                          p                      k                                              r                        ,                        y                                                                                            ⁢                                  |                                      p                    =                                          p                      0                                                                                                                                                                ⅆ                                                                  H                        1                                            ⁡                                              (                        p                        )                                                                                                  ⅆ                                          p                      k                                              r                        ,                        z                                                                                            ⁢                                  |                                      p                    =                                          p                      0                                                                                                                                                                ⅆ                                                                  H                        1                                            ⁡                                              (                        p                        )                                                                                                  ⅆ                                          τ                      k                      r                                                                      ⁢                                  |                                      p                    =                                          p                      0                                                                                                                              ⋮                                      ⋮                                      ⋮                                      ⋮                                                                                                                ⅆ                                                                  H                        j                                            ⁡                                              (                        p                        )                                                                                                  ⅆ                                          p                      k                                              r                        ,                        x                                                                                            ⁢                                  |                                      p                    =                                          p                      0                                                                                                                                                                ⅆ                                                                  H                        j                                            ⁡                                              (                        p                        )                                                                                                  ⅆ                                          p                      k                                              r                        ,                        y                                                                                            ⁢                                  |                                      p                    =                                          p                      0                                                                                                                                                                ⅆ                                                                  H                        j                                            ⁡                                              (                        p                        )                                                                                                  ⅆ                                          p                      k                                              r                        ,                        z                                                                                            ⁢                                  |                                      p                    =                                          p                      0                                                                                                                                                                ⅆ                                                                  H                        j                                            ⁡                                              (                        p                        )                                                                                                  ⅆ                                          τ                      k                      r                                                                      ⁢                                  |                                      p                    =                                          p                      0                                                                                                                              ⋮                                      ⋮                                      ⋮                                      ⋮                                                                                                                ⅆ                                                                  H                        M                                            ⁡                                              (                        p                        )                                                                                                  ⅆ                                          p                      k                                              r                        ,                        x                                                                                            ⁢                                  |                                      p                    =                                          p                      0                                                                                                                                                                ⅆ                                                                  H                        M                                            ⁡                                              (                        p                        )                                                                                                  ⅆ                                          p                      k                                              r                        ,                        y                                                                                            ⁢                                  |                                      p                    =                                          p                      0                                                                                                                                                                ⅆ                                                                  H                        M                                            ⁡                                              (                        p                        )                                                                                                  ⅆ                                          p                      k                                              r                        ,                        z                                                                                            ⁢                                  |                                      p                    =                                          p                      0                                                                                                                                                                ⅆ                                                                  H                        M                                            ⁡                                              (                        p                        )                                                                                                  ⅆ                                          τ                      k                      r                                                                      ⁢                                  |                                      p                    =                                          p                      0                                                                                                          )                    ︸                  H          ⁡                      (                          p              0                        )                                ⁢                  (                                                            Δ                ⁢                                                                  ⁢                                  p                  k                                      r                    ,                    x                                                                                                                          Δ                ⁢                                                                  ⁢                                  p                  k                                      r                    ,                    y                                                                                                                          Δ                ⁢                                                                  ⁢                                  p                  k                                      r                    ,                    z                                                                                                                          Δ                ⁢                                                                  ⁢                                  τ                  k                  r                                                                    )                    ︸                  Δ          ⁢                                          ⁢          p                      =            (                                                                  τ                                  1                  ,                  1                  ,                  k                                            +                              ɛ                                  1                  ,                  k                                            -                                                H                  1                                ⁡                                  (                                      p                    0                                    )                                                                                          ⋮                                                                              τ                                  1                  ,                  j                  ,                  k                                            +                              ɛ                                  j                  ,                  k                                            -                                                H                  j                                ⁡                                  (                                      p                    0                                    )                                                                                          ⋮                                                                              τ                                  1                  ,                  M                  ,                  k                                            +                              ɛ                                  M                  ,                  k                                            -                                                H                  M                                ⁡                                  (                                      p                    0                                    )                                                                        )              ︸      d      
If the individual propagation time estimates have different variances, the individual propagation time estimates may be weighted differently according to their variance when the position solution is calculated. This is achieved with the help of the following weighting matrix of a diagonal structure:
  W  =      (                                        σ            1                          -              2                                                0                          …                          …                          0                                      0                          ⋱                          ⋱                                                                          ⋮                                      ⋮                          ⋱                                      σ            i                          -              2                                                ⋱                          ⋮                                      ⋮                                                                          ⋱                          ⋱                          0                                      0                          …                          …                          0                                      σ            N                          -              2                                            )  
The iterative solution is then obtained in the following manner:
defining p0 (according to previous knowledge)for i = 1:N_iterations  Δp = [HT(p0)WH(p0)]−1HT(p0)Wd  p0 = p0 + Δpend  p = p0.
In this case, the convergence of the solution is generally achieved already after a few iterations. For an overdetermined system of equations with more than four sources, the solution obtained is the so-called “mean least squares” solution that minimizes the mean square error.
For a continuous estimation of the propagation time of incoming signals, which are variable in time due to the movement of a satellite and a receiver, a navigation receiver usually uses a combination of two control loops supporting each other. The so-called phase-lock-loop (PLL) for the control of the carrier phase is used to guarantee coherence with the received signal and to allow for a representation in the baseband. The so-called delay-lock-loop (DLL), illustrated in FIG. 2, synchronizes the received baseband signal with a locally simulated reference signal, formed in a replica generator, by maximizing their cross-correlation. A constant tracking of the maximum for the maintenance of the synchronization is achieved by corresponding shiftings (“early”, “late”) of the reference signal from which the propagation time of the signal from the satellite to the receiver can be determined.
The receiver will then determine its own position from the propagation times of at least four satellites. Reference is made in this context to FIG. 3, where a plurality of Delay Lock Loops DLL 1, . . . , DLL i, . . . , DLL N is used to estimate propagation time. From the propagation time estimates, an estimation of the position and of the clock error is obtained by solving corresponding equations.
In practice, this combination of DLL and PLL proves to be a robust realization of an almost optimal propagation time estimation device when there is no multipath propagation of the signals. If, however, the received signal is formed by a superposition of individual paths, which mainly occurs due to the transmitted signal being reflected or diffracted from objects in the vicinity of the receiver, the DLL will provide an erroneous estimation, which has an immediate effect on the precision of the position result.
Of the known signal processing methods for reducing the multipath error, most are based on more or less immediate modifications of the conventional DLL, aiming at reducing the influence of the additional paths as much as possible, i.e. to suppress this influence, as it were.
Besides the presumably most simple variant, the so-called “narrow correlator” [A. van Dierendonck, P. Fenton, T. Ford: “Theory and Performance of Narrow Correlator Spacing in a GPS Receiver” in Proceedings of the ION National Technical Meeting 1992, San Diego, Calif., USA, 1992], wide-spread use is also made, for example, of the so-called “strobe correlator” [L. Garin, F. van Diggelen, J. Rousseau: “Strobe and Edge Correlator Multipath Mitigation for Code” in Proceedings of the ION GPS 1996, Kansas City, Mo., USA, 1996], the so-called “gated correlator” [G. MacGraw, M. Brasch: “GNSS Multipath Mitigation using Gated and High Resolution Correlator Concepts” in Proceedings of the ION National Technical Meeting 1999, San Diego, Calif., USA, 1999] and the so-called “pulse-aperture correlator” [J. Jones, P. Fenton, B. Smith: “Theory and Performance of the Pulse Aperture Correlator” in NovAtel Technical Report, NovAtel Inc., Calgary, Alberta, Canada, 2004].
Another approach to the reduction of multipath errors is to include the additional paths in the formulation of the estimation problem and to solve the same by optimum methods or simplifications of such methods. These include the various variants of maximum likelihood (ML) estimation methods. Examples thereof are known from:
D. van Nee, J. Siereveld, P. Fenton, and B. Townsend: “The Multipath Estimating Delay Lock Loop: Approaching Theoretical Accuracy Limits” in Proceedings of the IEEE Position, Location and Navigation Symposium 1994, Las Vegas, Nev., USA, 1994;
L. Weill: “Achieving Theoretical Accuracy Limits for Pseudo-ranging in the Presence of Multipath” in Proceedings of the ION GPS 1995, Palm Springs, Calif., USA, 1995;
J. Selva Vera: “Complexity Reduction in the Parametric Estimation of Superimposed Signal Replicas” in Signal Processing, Elsevier Science, vol. 84, no. 12, pages 2325-2343, 2004; and
P. Fenton, J. Jones: “The Theory and Performance of Novatel Inc's Vision Correlator” in Proceedings of the ION GNSS 2005, Long Beach, Calif., USA, September. 2005.
While all these known estimation methods are based on the same concept, they differ in the details of the manner in which they strive to realize the solution as effectively as possible. In effect, an immediate implementation will fail due to the unrealistically high complexity. With static channels, the ML estimator is optimal and obtains clearly better results than other methods, especially when the additional paths only show slight relative delays.
An estimator for multipath situations that is based on sequential importance sampling (SIS) methods (particle filtering) and is considered in an article by P. Closas, C. Fernandez-Prades, J. Fernandez-Rubio, A. Ramirez-Gonzalez: “Multipath Mitigation using Particle Filtering” in Proceedings of the ION GNSS 2006, Fort Worth, Tex., USA, September 2006, is advantageous in that it allows the additional use of a-priori knowledge about the channel properties. Further, the instantaneous solution and its covariance matrix are used to estimate the subsequent point in time, whereby the time-related correlation of the estimation parameters is taken into account.
One possible way to combine the knowledge about the temporal correlation of the estimation parameters with the methods for an efficient implementation of an ML estimator, proposed in the above mentioned article by J. Selva Vera, is presented in an article by B. Krach, M. Lentmaier: “Efficient Soft-Output GNSS Signal Parameter Estimation using Signal Compression Techniques” in Proceedings of the 3rd ESA Workshop on Satellite Navigation User Equipment Technologies, Navitec 2006, Noordwijk, The Netherlands, December 2006.
The knowledge about the parameter development may be provided, for example, by the DLL/PLL loop of a conventional receiver. The method allows to calculate the a-posteriori distribution of the estimation parameters whose maximum can be determined, for example, with the methods stated in the above mentioned article by J. Selva Vera.
Further approaches known from A. Giremus and Y.-Y. Tourneret: “Joint detection/estimation of multipath effects for the global positioning system”, Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, 2005, vol. 4, Philadelphia, Pa., USA, March 2005, pages 14-20, and P. Closas, C. Fernandez-Prades, and J. Fernandez-Rubio: “Maximum likelihood estimation of position in GNSS”, IEEE Signal Processing Letters, vol. 14, no. 5, pages 359-362, May 2007, use the binding of the different propagation time measurements via the position parameters to suppress or detect the multipath errors.
All presently existing methods are suboptimal and/or not adapted to the dynamic properties of the channels. In environments that are critical with respect to multipath propagation, such as in urban canyons in cities, the navigation receivers currently known do not work reliably.
When implementing a maximum likelihood (ML) estimation method, the number of paths is presumed to be known. In practice, this number has to be estimated or assumed, however, whereby the performance of these methods can be much impaired by erroneous assumptions or erroneous estimations of this parameter.